# Questions tagged [metric-embeddings]

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### Hyperbolic space embeds into Wasserstein space

Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B_{\mathbb{H}^n}(x,r)$. Are ...
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### Explicit formula for embedding Cayley graph of free group into hyperbolic space

The problem is to embed Cayley graph of free group with $n\geq2$ generators (the same as Bethe lattice with coordination number $2n$) into any model of $\mathbb{H}^2$ (we have no model preference, the ...
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Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with $$\left\{B\left(x_k,\frac1{n}\right)... 10 votes 2 answers 551 views ### Are hyperbolic spaces actually better for embedding trees than Euclidean spaces? There is a folklore in the empirical computer-science literature that, given a tree (X,d), one can find a bi-Lipschitz embedding into a hyperbolic space \mathbb{H}^n and that n is "much ... 1 vote 0 answers 64 views ### Best estimate on doubling constant of a finite metric space Let (X,d) be a finite metric space. Clearly, (X,d) is a doubling metric space but is there a 'best' estimate of (X,d)'s doubling constant? Probability based on its cardinality, diameter, and ... 4 votes 2 answers 157 views ### Rozendorn's Article I'm researching the isometric dips of the hyperbolic plane and in particular I'm interested in reading the results of Rozendorn who proved that the hyperbolic plane is isometrically immersed in \... 2 votes 0 answers 81 views ### Almost Lipschitz embedding of compact metric measure spaces into Euclidean spaces Let (X,d) be a compact metric space, m be a metric outer-measure on X. Are there 'mild conditions' on X ensuring the existence of a positive integer N\geq 3 such that there exist x_1,\dots,... 7 votes 1 answer 177 views ### Does there exist a countable metric space which is Lipschitz universal for all countable metric spaces? Is there a countable metric space U such that any countable metric space is bi-Lipschitz equivalent to a subset of U? How about c_{00}(\mathbb{Q}) where \mathbb{Q} is the rational numbers? ... 1 vote 0 answers 63 views ### intuition about Gaussian processes over a finite space In a paper that I am reading the authors defines \mathbb P(n,q) the space of covariance tensors for \mathbb R^q-valued Gaussian processes on an abstract finite space K=\{x_1,\dots,x_n\}. In his ... 2 votes 0 answers 62 views ### Kernels with finite dimensional feature spaces Suppose x,y \in \mathbb{R}^n for some given fixed n. Consider a kernel K(x,y) = f(\langle x, y \rangle), I'd like to know which functions f admit a finite dimensional feature map. In other words,... 5 votes 1 answer 280 views ### When is a metric space a snowflake? Let (X,d) be a metric space. For any 0<\epsilon<1, we call the metric space (X,d^{\epsilon}); where d^{\epsilon}(x,y)\triangleq (d(x,y))^{\epsilon} the \epsilon-snowflake of (X,d). ... 6 votes 1 answer 228 views ### Why (\mathrm{Lip}([0,1]^2))^* is finitely representable in 1-Wasserstein space over the plane? In "Snowflake universality of Wasserstein spaces"" by Alexandr Andoni, Assaf Naor, and Ofer Neiman, they have the following notation: For a metric space X they write \mathcal{P}_1(X) ... 4 votes 1 answer 170 views ### Bi-Hölder embeddings of finite metric spaces This is a reference request. There is a large body of work, I'm familiar with, that describes the existence of bi-Hölder embeddings of finite metric spaces into Euclidean space (such as this ... 5 votes 0 answers 123 views ### Correspondence between Riemannian metrics and Euclidean embeddings Given a sufficiently smooth manifold M, a Riemannian metric on M induces an isometric embedding into Euclidean space by Nash's theorem, (non-canonically, non-uniquely) an embedding of M into ... 3 votes 1 answer 122 views ### Banach embedding of finite dimensional spaces Recall that: let 0<r<s<2, then \ell_r uniformly contains a subspace isomorphic to \ell_s^m, m\ge 1 (see [JS]). I am wondering whether are any result for the case when r>s>2? ... 3 votes 0 answers 87 views ### An explicit (maybe algebraic) isometric embedding of the double torus with constant curvature K = -1 The following question is related to this previous question, Canonical immersion of the double torus: Is there any known explicit (maybe algebraic) isometric embedding of a genus 2 surface endowed ... 2 votes 0 answers 73 views ### Dense embeddings into Euclidean space The question is a follow-up on this old post. Fix a positive integer d and consider \mathbb{R}^d with its usual Euclidean topology. Given a metric space (X,\delta_X), what conditions are ... 10 votes 0 answers 755 views ### Two questions around the abc-conjecture Let d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}, d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)} be two metrics on natural numbers. The abc-conjecture can be formulated using these two metrics as: For ... 2 votes 0 answers 109 views ### Is the matrix \mu_f(X_i \cap X_j) positive definite? Let X_1,\ldots, X_n be finite subsets of some larger finite set Z. Let f:Z \rightarrow \mathbb{R}_{>0} be any function, and define a (counting) measure \mu_f(X) = \sum_{x \in X} f(x) for a ... 2 votes 0 answers 107 views ### Partitions of unity with arbitrary Lip-constants Lets make things simple. Suppose we have a compact metric space (X,d) and then some Lipschitz partition of unity exists, say a collection \mathcal{F}=\{f_n\} subordinate to some open cover \... 2 votes 0 answers 111 views ### Embedding a binary subspace to l_2 in a much lower dimension I'm trying to find a way to embed a binary linear subspace of dimension n (a linear code) to the Euclidian space while reducing the dimension significantly. The subspace (or code) contains points ... 5 votes 0 answers 209 views ### Is this function embeddable in Euclidean space? Let X = \{v_1,\ldots,v_n\} be a set of vectors non-zero vectors v_i \ge 0 and such that the vectors are pairwise linear independent. Define a function on this set X:$$d(v,w) = 1-\frac{2 \... 2k views

### Trigonometry / Euclidean Geometry for natural numbers?

Let $d(a,b) = 1 - \frac{2\gcd(a,b)^3}{ab(a+b)}$ be a metric on natural numbers without $0$. The metric space $X = \{x_0,x_1,\cdots,x_n\},n>2$ is isometric embeddable in $\mathbb{R}^n$ if and only ... 100 views

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### Green's Function for Fractional Laplacian on the Union of Two Balls

I have two disjoint open intervals $B_1, B_2 \subset \mathbb{R}$, and variables $0 < s < 1$ and $t \in B_1 \cup B_2$. I want to solve: $$r_{B_1 \cup B_2}(\Delta^{s} f) = \delta_t$$ for $f$. ...
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### Isometric embedding of a genus g surface

Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $\mathbb{R}^4?$
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### Suppose that a metric space allows David–Semmes regular map to some Euclidean space. Does it allow bi-Lipschitz embedding into some Euclidean space?

I want to ask about the progress on Question 8 from "Thirty-three yes or no questions about mappings, measures, and metrics" by Juha Heinonen and Stephen Semmes. Is it still open? If yes, ...
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### Inclusion of convex polytopes and embedding from $\ell_2$ to $\ell_\infty$

I would like to dig deeper into the problem posted Probability that a convex shape contains the unit ball: If you pick n points uniformly at random from the surface of a d dimensional sphere of ...
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### Do manifolds with non-negative Ricci curvature allow bi-Lipschitz embeddings into Euclidean spaces?

QUESTION: Let $n$ be a natural number. Is it true that there exist $N(n), D(n) > 0$ such that any complete $n$-dimensional Riemannian manifold of nonnegative Ricci curvature can be embedded into $N$...
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### Embedding Turing machine [closed]

I have some questions about Turing machines. Is there an embedding method where you embed Turing machines, finite automata into continuous space or graphs? Or are there geometrical approaches to ...
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### Johnson-Lindenstrauss Lemma on $S^{d-1}$

Consider the Johnson-Lindenstrauss lemma in the case where we can assume the $n$ input points $x_i$ in $\mathbb{R}^d$ are actually located on the sphere S^{d-1}(r):=\{u=(u_1,\ldots,u_{d}): u_1^2+\...
1 vote
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### Expectation of a combinatorial extremal random variable?

Consider the finite set $\chi(D)$ of all sets of integer points in $\Bbb Z^n$ around origin which have distance at most $D$ from each other and pick a set $\mathcal P(D)$ from set of sets $\chi(D)$ ...
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### Totally distance non-preserving transformations

JL lemma (https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma) guarantees if you have a set of $K$ points in $\Bbb R^N$ a random transformation guarantees that the set can be projected ...
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### Finitely isometrically persistent metric spaces

The goal of this question is to develop further the discussion initiated in Under which conditions is it possible to find points with same distances under bi-Lipschitz map. The mentioned question was ...
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### Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?

Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$? It seems to me that it is an interesting ...
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### Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$? There are some results of the ...
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### Embedding graphs into hyperbolic spaces

Do we know of a characterization as to when does a graph have a "good" embedding into a hyperbolic space? (And does having such an embedding have a spectral or wavelet analysis signature?) I don't ...
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### Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?

Question 1. Does every CAT(0) space embed isometrically inside an integral of $\mathbb{R}$-trees? Here an integral of $\mathbb{R}$ trees means the set of functions from a measure space $\mathcal{F}$ ...
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### Canonical immersion of the double torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...
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### Embedding Euclidean buildings into products of trees

A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.) Question: Is it true ...
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### Embeddings of finitely generated groups into uniformly convex Banach spaces

de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if ...
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### Reference request: embedding the hyperbolic triangulation in $\mathbb{R}^3$

Let $T_d$ be the infinite valence $d$ triangulation of the hyperbolic plane, where each triangle is equilateral and $d \ge 7$. Question: Is there an isometric embedding from $T_d \to \mathbb{R}^3$? ...
Consider a graph $G$ with nonnegative edge weights. Question: In $\mathbb{R}^3$, how hard is it to assign coordinates to vertices such that the Euclidean length of each edge is equal to its weight? ...